regular representation (original) (raw)

Given a group G, the regular representation of G over a field K is the representation ρ:G⟶GL⁡(KG) whose underlying vector space KG is the K–vector space of formal linear combinations of elements of G, defined by

for ki∈K, g,gi∈G.

Equivalently, the regular representation is the induced representation on G of the trivial representation on the subgroup {1} of G.